I believe what the saying means is this: If a theory X is accepted as true, then that implies that some amount of reasonable evidence exists in support of X. If we are to prove ~X, then we must show not only evidence in favor of ~X but also explain why the evidence in favor of ~X is stronger than the evidence in favor of X (e.g. using better instruments results in more precise measurements). The new evidence must be more "extraordinary" in the sense that it must be strong enough to overturn the evidence in support of the "ordinary claim".
The line "extraordinary claims require extraordinary evidence" is just more poetic than the paragraph above.
The line "extraordinary claims require extraordinary evidence" is just more poetic than the paragraph above.